Phonon Anomalies Induced by Superconductivity

We calculate the electronic polarizability in the superconducting state near extremum vectors ~ Q0 of the Fermi surface. A pole appears in the polarizability at frequencies ω near the superconducting gap 2∆ which leads to a sharp peak at ω just below 2∆ in the lattice vibrations near ~ Q0. A second order transition to a charge density wave state below the superconducting transition is shown to be unlikely. The results are compared with the recent inelastic neutron scattering measurements in compounds, ANi2B2C, A = Y or Lu. 74.20.-z,74.25.kc Typeset using REVTEX 1 Strong momentum dependence of the electronic polarizability leads to characteristic features in the dispersion relation of phonons in simple metals–the Kohn anomalies. [1] Transition metals and compounds are particularly beset with wiggles and dips in their phonon dispersions. Strong electron-phonon interactions and local field effects are responsible together with nesting features in the band structure for the anomalies and for the occurrence of charge density wave transitions [2]. It is well known that the leading order polarizability Π0 ( ~ Q, ω), Fig. 1 (a), of an electron gas in d-dimensions for ~ Q = 2~kF for a spherical Fermi-surface or ellipsoidal Fermi-surface across the major and minor axes of the Fermi-surface in the normal state is [3] ReΠ0 (2 ~kF , ω) ≃ N(0)[1 + (N(0)ω)ln( EF ω )]. (1) We show the existence of singularities in the polarizability at ω ≃ 2∆ in s-wave superconductors near the extremum vectors of the Fermi-surface which are stronger than those of the normal state at ω → 0. By an extremum vector we mean ~ Q0 such that ǫ ~ Q0/2 and ǫ− ~ Q0/2 are both on the Fermi-surface and for small deviations |~k| << | ~ Q0| ~k− ~ Q0 2 = −vF‖~k‖ + ∑ i ~k2 i⊥ 2mi⊥ ǫ ~k+ ~ Q0 2 = vF‖~k‖ + ∑